Mathematische Formelsammlung

14.2 Streuungsmaße s 2 = s​ _  x ​ 2 = σ 2 … empirische Varianz; s = s​ _  x ​= σ … empirische Standardabweichung; d​ _  x ​… mittlere lineare Abweichung; s 2 = s​ _  x ​ 2 = σ 2 = ​  (x 1 – ​ _ x​) 2 + (x 2 – ​ _ x​) 2 + … + (x n – ​ _ x​) 2 ___  n ​= ​  1 _ n ​​ ;  i = 1 ​  n ​ (x i – ​ _ x​) 2 ​ s = ​ 9 __ s 2 ​= s​ _  x ​= ​ 9 __ s​ _  x ​ 2 ​= σ = ​ 9 __ σ 2  ​= ​ 9 ________________ ​  (x 1 – ​ _ x​) 2 + (x 2 – ​ _ x​) 2 + … + (x n – ​ _ x​) 2 ___  n ​​= ​ 9 _______ ​  1 _ n ​​ ;  i = 1 ​  n ​ (x i – ​ _ x​) 2 ​ ​ Andere Definition s 2 = ​  (x 1 – ​ _ x​) 2 + (x 2 – ​ _ x​) 2 + … + (x n – ​ _ x​) 2 ___  n – 1 ​= ​  1 _  n – 1 ​​ ;  i = 1 ​  n ​ (x i – ​ _ x​) 2 ​ s = ​ 9 __ s 2  ​ Bei gegebenen absoluten Häufigkeiten: s 2 = ​  (a 1 – ​ _ x​) 2  H 1 + (a 2 – ​ _ x​) 2  H 2 + … + (a k – ​ _ x​) 2  H k ___  n ​= ​  1 _ n ​​ ;  i = 1 ​  k ​ (a i – ​ _ x​) 2 ​H i = ​  1 _ n ​​ ;  i = 1 ​  k ​ (a i – ​ _ x​) 2 H n  (a i )​ bzw.  σ 2 = ​  (x 1 – ​ _ x​) 2  h 1 + (x 2 – ​ _ x​) 2  h 2 + … + (x m – ​ _ x​) 2  h m ___  n ​= ​  1 _ n ​​ ;  i = 1 ​  m ​ (x i – ​ _ x​) 2  h i ​ Andere Definition s 2 = ​  (x 1 – ​ _ x​) 2  h 1 + (x 2 – ​ _ x​) 2  h 2 + … + (x m – ​ _ x​) 2  h m ___  n – 1 ​= ​  1 _  n – 1 ​​ ;  i = 1 ​  m ​ (x i – ​ _ x​) 2  h i ​ Bei gegebenen relativen Häufigkeiten: s 2 = (a 1 – ​ _ x​) 2  h n  (a 1 ) + (a 2 – ​ _ x​) 2  h n  (a 2 ) + … + (a k – ​ _ x​) 2  h n  (a k ) = ​ ;  i = 1 ​  k ​ (a i – ​ _ x​) 2  h n  (a i )​ bzw.  σ 2 = (x 1 – ​ _ x​) 2  r 1 + (x 2 – ​ _ x​) 2  r 2 + … + (x m – ​ _ x​) 2  r m = ​ ;  i = 1 ​  m ​ (x i – ​ _ x​) 2  r i ​ Verschiebungssatz: s 2 = s​ _  x ​ 2 = σ 2 = ​  1 _ n ​(x 1 2 + x 2 2 + … + x n 2 ) – ​ _ x​ 2 = ​  1 _ n ​ ​ (   ​ ;  i = 1 ​  n ​ x i 2 ​  ) ​– ​ _ x​ 2 s 2 = ​  a 1 2  H n  (a 1 ) + a 2 2  H n  (a 2 ) + … + a k 2  H n  (a k ) ___  n ​– ​ _ x​ 2 s 2 = ​ [ a 1 2  h n  (a 1 ) + a 2 2  h n  (a 2 ) + … + a k 2  h n  (a k ) ] ​– ​ _ x​ 2 Statistik 35 Nur zu Prüfzwecken – Eigentum des Verlags öbv